where the index \(i\)  represents the nodes of the mesh inside the domain \(\mathcal {B}\) , \(j \in N(i)\)  represents the neighbor nodes of the node \(i\) , including itself. The term \(\hat{u}_j\)  represents the output of the neural network at the nodal locations \(j\)  that depends on trainable parameters \(\mbox{$\theta $}{}\) . For traditional PINNs, we use automatic differentiation to compute \(\Delta \hat{u}\)  and construct the loss term \(MSE_{\rm PDE}\) . We use 50 eigenfunctions to represent the domain \(\mathcal {B}\)  for \(\Delta \) -PINNs, and 3 hidden layers of 100 neurons for all cases. We train the network with ADAM [1]} for 50,000 iterations using a batch size of 30 samples to evaluate each of the terms of the loss function. We use 361 values of \(u\)  located at the boundaries as training data, as shown in Figure REF . We note that ewe use the same operator, the Laplacian, for the PDE that governs this problem and to construct the input of the neural network. However, the eigenfunctions are computed with homogeneous Neumann boundary conditions, which guarantees that the eigenfunctions \(v_i\)  are not solutions to the boundary value problem shown in REF  since they have different boundary conditions.
